Allawi led his Iraqiya slate to a narrow victory in March 7 elections, though fell well short of the 163-seat majority needed to form a government. Maliki, meanwhile, won support from the alliance loyal to anti-American cleric Moqtada Sadr, putting Maliki within reach of the majority.
Allawi suggested to The Wall Street Journal that he would agree to power-sharing talks with Maliki provided all top government positions, including prime minister, were up for debate.
Adnan al-Sarraj, a Maliki supporter, responded by saying the position of prime minister was not a matter for negotiation.
Allawi added that he wanted to see plans on the table for resolving internal border issues and lingering oil revenue concerns.
A spokeswoman for Iraqiya said in a separate statement that the secular slate welcomed a proposal by Kurdish President Massoud Barzani to appoint representatives from leading parties to move negotiations forward.
Meanwhile, an Iraqi official told London's pan-Arab daily Asharq al-Awsat on condition of anonymity that there was the potential for a coup in Iraq.
"We do not completely exclude a military coup, since Iraq's political history is replete with military coups," the source said.
2014: The Year in Music [PHOTOS]